Reliability analysis of structures with multimodal distributions based on direct probability integral method

For practical structures, some input random variables follow multimodal distributions, and conventional reliability analysis methods may result in large computational errors. In this paper, a novel direct probability integral method (DPIM), which decouples governing equation of structure and the probability density integral equation (PDIE), is proposed to address the static and dynamic reliability assessment of structures involving random variables with multimodal distributions. Firstly, the multimodal probability density functions (PDFs) of input random variables are established by the Gaussian mixture model. Then, using numerical integration and smoothing technique of Dirac delta function, the PDFs of structural responses with multimodal random variables are achieved by solving the PDIE, in which three numerical integration algorithms, namely, the Quasi-Monte Carlo approach, the sparse grid approach, and Generalized F-discrepancy-based point selection approach are employed. Further, the reliability of static structure can be readily obtained by integrating the PDF of response function, while the dynamic reliability can be evaluated by DPIM combining with extreme value distribution of stochastic process. Finally, several examples demonstrate the superiority of DPIM, and the Generalized F-discrepancy-based point selection approach has the highest accuracy and efficiency for solving PDIE. The characteristics of uncertainty propagation in structures involving multimodal distributions of input random variables are revealed.